five_pow_4
plain-language theorem explainer
Five to the fourth power equals 625 is recorded as a foundational arithmetic identity within the product recognition lattice. Cross-domain researchers in Recognition Science cite this when constructing four-fold state spaces such as cognitive times tumor-stage configurations. The verification proceeds by a direct decision tactic that computes the natural number power exactly.
Claim. $5^{4} = 625$
background
The Product Recognition Lattice enumerates sizes of cross-domain recognition state spaces via successive powers of five. The module documentation lists 5^4 = 625 as the four-fold level, for example the product of a cognitive domain with a tumor-stage domain. This hierarchy begins at 5^2 = 25 for pairs of D=5 domains and continues through 5^3 = 125 for single triples up to the joint cognitive-oncology space of size 5^6 = 15625, which is required to satisfy the bound 5^6 < 2^14.
proof idea
The proof is a one-line wrapper that invokes the decide tactic to evaluate the arithmetic expression (5 : ℕ)^4 and confirm it equals 625.
why it matters
This declaration supplies the concrete numerical value for the four-fold entry in the lattice hierarchy. It supports the module's structural claim that cross-domain products produce bounded recognition state spaces, including the information-theoretic constraint that the joint cognitive-oncology configuration fits inside 14 bits. In the Recognition Science framework it anchors the enumeration of product spaces that arise when combining domains under the recognition composition law.
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